A Linear Bound on the Complexity of the Delaunay
triangulation of points on polyhedral surfaces
Dominique Attali, Jean-Daniel Boissonnat
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Dominique Attali, Jean-Daniel Boissonnat. A Linear Bound on the Complexity of the Delaunay
triangulation of points on polyhedral surfaces. RR-4453, INRIA. 2002. <inria-00072135>
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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
A Linear Bound on the Complexity of the Delaunay
triangulation of points on polyhedral surfaces
Dominique Attali — Jean-Daniel Boissonnat
N° 4453
Avril 2002
ISSN 0249-6399
ISRN INRIA/RR--4453--FR+ENG
THÈME 2
apport
de recherche
A Linear Bound on the Complexity of the Delaunay
triangulation of points on polyhedral surfaces
Dominique Attali , Jean-Daniel Boissonnat
Thème 2 — Génie logiciel
et calcul symbolique
Projet Prisme
Rapport de recherche n° 4453 — Avril 2002 — 20 pages
Abstract: Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction.
Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on
the complexity of the Delaunay triangulation of such point sets.
Although the Delaunay triangulation of points in can be quadratic in the worst-case, we show
that, under some mild sampling condition, the complexity of the 3D Delaunay triangulation of points
distributed on a fixed number of facets of (e.g. the facets of a polyhedron) is linear. Our bound
is deterministic and the constants are explicitly given.
Key-words: Computational geometry, Delaunay triangulation, Voronoi diagram, polyhedral surfaces, complexity, surface sampling, surface modeling, surface reconstruction
This work has been partially supported by the ARC Costic of INRIA and by the ECG Project (Effective Computational
Geometry for Curves and Surfaces) of the IST Programme of the EU (contract No IST-2000-26473).
LIS, ENSIEG, Domaine Universitaire, BP 46, 38402 Saint Martin d’Hères Cedex. e-mail: [email protected]
INRIA, 2004 Route des Lucioles, BP 93, 06904 Sophia-Antipolis, France.
e-mail :
[email protected]
Unité de recherche INRIA Sophia Antipolis
2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)
Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65
Une borne linéaire sur la complexité de la triangulation de
Delaunay de points distribués sur des surfaces polyédriques
Résumé : Les triangulations de Delaunay et les diagrammes de Voronoi trouvent des applications
nombreuses en modélisation de surfaces : génération de maillages, déformation, reconstruction de
surfaces. Beaucoup d’algorithmes dans ce contexte commencent par construire la triangulation de
Delaunay d’un ensemble fini de points pris sur la surface. Leur complexité dépend donc de la
complexité de la triangulation de Delaunay des points.
, c’est-à-dire le nombre de ses
La complexité de la triangulation de Delaunay de points de
faces, peut être
. En particulier, dans , le nombre de tétraèdres peut être quadratique.
Dans cet article, nous donnons une borne linéaire sur la complexité de la triangulation de Delaunay
de points distribués sur un nombre fixé de facettes de , par exemple les faces d’un polyèdre, et
vérifiant une hypothèse d’échantillonnage uniforme assez faible. Notre borne est déterministe et les
constantes sont données explicitement.
Mots-clés : Géométrie algorithmique, triangulation de Delaunay, diagramme de Voronoi, surfaces
polyédriques, complexité, échantillonnage de surface, modélisation de surfaces, reconstruction de
surfaces
Complexity of Delaunay triangulations
3
Delaunay triangulations and Voronoi diagrams have found numerous applications in surface
modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many
algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the
complexity of the Delaunay triangulation of such point sets.
Although the complexity of the Delaunay triangulation of points in may be quadratic in the
worst-case, we show in this paper that it is only linear when the points are distributed on a fixed
number of well-sampled facets of (e.g. the facets of a polyhedron). Our bound is deterministic
and the constants are explicitly given.
1 Introduction
Delaunay triangulations and Voronoi diagrams are among the most thoroughly studied geometric
data structures in computational geometry. Recently, they have found many applications in surface
modeling, surface mesh generation [13], deformable surface modeling [23, 17], medial axis approximation [4, 10, 22], and surface reconstruction [3, 1, 9, 2, 7, 6]. Many algorithms in these applications
begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered
over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets.
It is well known that the complexity of the Delaunay triangulation of points in , i.e. the
number of its simplices, can be
[11]. In particular, in , the number of tetrahedra can
be quadratic. This is prohibitive for applications where the number of points is in the millions,
which is routine nowadays. Although it has been observed experimentally that the complexity of
the Delaunay triangulation of well-sampled surfaces is linear (see e.g. [9, 14]), no result close to
this bound has been obtained yet. Our goal is to exhibit practical geometric constraints that imply
subquadratic and ultimately linear Delaunay triangulations. Since output-sensitive algorithms are
known for computing Delaunay triangulations [12], better bounds on the complexity of the Delaunay
triangulation would immediately imply improved bounds on the time complexity of computing the
Delaunay triangulation.
First results on Delaunay triangulations with low complexity have been obtained by Dwyer [15,
16] who proved that, if the points are uniformly distributed in a ball, the expected complexity of
the Delaunay triangulation is only linear. Recently, Erickson [18, 19] investigated the complexity
of three-dimensional Delaunay triangulations in terms of a geometric parameter called the spread,
which is the ratio between the largest and the smallest interpoint distances. He proved that the
complexity of the Delaunay triangulation of any set of points in with spread is .
Despite its practical importance, the case of points distributed on a surface has not received much
attention. A first result has been obtained by Golin and Na [20]. They proved that the expected complexity of 3D Delaunay triangulations of random points on any fixed convex polytope is . Very
recently, they extended their proof to the case of general polyhedral surfaces of and obtained a
bound on the expected complexity of the Delaunay triangulation [21]. Deterministic
bounds have also been obtained. Attali and Boissonnat [5] proved that, for any fixed polyhedral surface , any so-called “light-uniform -sample” of of size has only Delaunay tetrahedra.
RR n° 4453
4
Attali & Boissonnat
If the surface is convex, the bound reduces to . Applied to a fixed uniformly-sampled
surface, the result of Erickson mentioned above shows that the Delaunay triangulation has complex . This bound is tight in the worst-case. It should be noticed however that Erickson’s
ity definition of a uniform sample is rather restrictive and does not allow two points to be arbitrarily
close (in which case, the spread would become infinite).
In this paper, we consider the case of points distributed on a fixed number of planar facets in ,
e.g. the facets of a given polyhedron. Under a mild uniform sampling condition, we show that the
complexity of the Delaunay triangulation of the points is linear. Our bound is deterministic and the
constants are explicitly given.
2 Definitions and notations
2.1 Voronoi diagrams and Delaunay triangulations
be a set of points of . The Voronoi cell of is
!" $# %!&(' *),+ .-/0 1
where &!32 denotes the Euclidean distance between the two points 452 of . The collection
of Voronoi cells is called the Voronoi diagram of , denoted 6 87 9 . The Delaunay triangulation
of , denoted :<; is the dual complex of 6 87 (see Figure 1). If there is no sphere passing
through =?>A@ points of , :$; 9 is a simplicial complex that can be obtained from 6 87 9 as
follows. If CB is a subset of points of whose Voronoi cells have a non empty intersection, the
convex hull D0E GF HB is a Delaunay face and all Delaunay faces are obtained this way. It is well
known that the balls circumscribing the = -simplices in :<; cannot contain a point of in their
interior. The complexity of :<; is the number of its faces, which is also the number of faces of
the dual Voronoi diagram.
A ball or a disk is said to be empty iff its interior contains no point of . We also say that a
Let
sphere is empty if the associated ball is empty.
2.2 Notations
I
L
J I
M 4ON P 45N V W "V
K N
For a curve , we denote by
its length. For a portion of a surface , we denote by
its
area, and by
its boundary. We further denote by
(
) the ball (sphere) of radius
centered at , and by
the disk lying in plane centered at
and of radius .
Let
be a region of . The plane containing is called a supporting plane of . We
define:
YX QSR TUON N
V
V
[ Z R <\V Q R TU ] _a^ `(
[ b R <\ V QSR TU X Z R is obtained by growing by within its supporting plane V and b R is obtained by
shrinking by within its supporting plane[V Z . Whenc
the
b supporting plane is unique or when it is
V
clear from the context, we will simply note
and
.
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Complexity of Delaunay triangulations
5
Figure 1: Voronoi diagram of a set of points on the left and its dual Delaunay triangulation on the
right.
2.3 Polyhedral surfaces
We call polyhedral surface a finite collection of bounded polygons, any two of which are either
disjoint or meet in a common edge or vertex. The polygons are called facets. Notice that we allow
an arbitrary number of polygons to be glued along a common edge. In the mathematical literature,
such an object is called a pure two-dimensional piece-wise linear complex. We prefer to use in this
paper the term surface since surfaces are our primary concern.
In the rest of the paper, denotes an arbitrary but fixed polyhedral surface. Three quantities
, and will express the complexity of the surface :
denotes the number of facets of ,
its area, and the sum of the lengths of the boundaries of the facets of :
aaK J 9L 5 Observe that, if an edge is incident to facets, its length will be counted times.
We consider two zones on the surface, the -singular zone that surrounds the edges of and the
-regular zone obtained by shrinking the facets.
X
Definition 1 Let . The -regular zone of a facet consists of the points of at distance
greater than from the boundary of . The -regular zone of is the union of the -regular zones of
its facets. We call -singular zone of (resp. ) the set of points that do not belong to the -regular
zone of (resp. ).
Observe that the -regular zone of the facet is of the edges of .
2.4 Sample
X
b
. The -singular zone of consists exactly
Any finite subset of points
is called a sample of . The points of are called sample points.
We impose two conditions on samples. First, the facets of the surface must be uniformly sampled.
Second, the sample cannot be arbitrarily dense locally.
RR n° 4453
6
Attali & Boissonnat
X
W
• the ball M TU encloses at least one point of ]
• the ball M TU@ encloses at most points of ]
Definition 2 Let be a polyhedral surface.
facet of and every point
:
is said to be a
8 -sample of iff for every
,
.
The 2 factor in the second condition of the definition is not important and is just to make the
constant in our bound simpler. Any other constant and, in particular 1, will lead to a linear bound.
In the rest of the paper, denotes a -sample of and we provide asymptotic results when
the sampling density increases, i.e. when tends to . As already mentioned, we consider and the
surface (and, in particular, the three quantities , and ) to be fixed and not to depend on .
Several related sampling conditions have been proposed.
Amenta and Bern have introduced -samples [3] that fit locally the surface shape : the point density
is high where the surface has high curvature or where the object or its complement is thin. However
this definition is not appropriate for polyhedral surfaces since an -sample, as defined in [3], should
have infinitely many points.
Erickson has introduced a notion of uniform sample that is related to ours but forbids two points
to be too close [18]. Differently, our definition of a -sample does not impose any lower bound
on the minimal distance between two sample points.
In [5], Attali and Boissonnat use a slightly different definition of a -sample. They assumed
that for every point
, the ball
encloses at least one sample point and the ball
encloses sample points. With this sampling condition, they proved that the complexity of the
Delaunay triangulation is for general polyhedral surfaces and for convex polyhedral
surfaces. Our definition of a -sample is slightly more restrictive since the facets need to be
sampled independently of one another, which leads to add a few more sample points near the edges.
However, the two conditions are essentially the same and our linear bound holds also under the
slightly more general sampling condition of [5].
Golin and Na [20, 21] assume that the sample points are chosen uniformly at random on the
surface. The practical relevance of such a model is questionable since data are usually produced in
a deterministic way.
8 M T U
3 Preliminary results
X
X
designates a polyhedral surface and
number of sample points in the region
We first establish two propositions relating
Lemma 3 Let be a facet of . For any
K
#
X
8 M T 45N .]
a -sample
of . Let
be the
. Let
be the total number of sample points.
and . We start with the following lemma:
# K c Z
, we have:
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Complexity of Delaunay triangulations
7
[Z Figure 2: A maximal set of non-intersecting disks contained in
of obtained by doubling the radii of the disks.
4 Q &O
Z and the corresponding covering
*Z
Proof. Let be a maximal set of non-intersecting disks lying inside . Because
the set of disks is maximal, no other disk can be added without intersecting one of the disks
than from a point (see
. This implies that no point of is at distance greater
Figure 2). Therefore, is a packing of
and is a covering of .
Consequently:
Q &O
U Q "5
K #
RR n° 4453
@
Q
&
4 O @
Z
# K [Z
8
Attali & Boissonnat
WZ
Q O The disks lie in
. Therefore, the centers of the disks lie in . By assumption, the disk contains at least one sample point and the disk contains at most sample points. Hence :
Q WO @
# #
K #
and
Proposition 4 Let #
K
[Z , we have:
from below. Summing over the facets of , we get :
Proof. We first apply Lemma 3 to bound
We apply again Lemma 3 to bound
X
be a facet of . For any
# K c Z
#
(1)
# K c Z
from above.
Eliminating from the two inequalities yields the result.
I X be a curve contained in . Let . We have:
J TI TI Z # @ > - J I # @ 9@ > - Z can be covered by Proof. Arguing as in the proof of Lemma 3, we see that the region I
disks of radius @ centered on I and contained in the supporting plane of .
Applying Lemma 3 to a disk with radius @ , we get:
# 9@ > 9@ > - Therefore, we have :
TI Z # @ > - J I Proposition 5 Let be a facet of . Let
From Equation 1, we get:
- # @ Combining the two inequalities leads to the result.
INRIA
Complexity of Delaunay triangulations
9
Q R 2
2 D
P*] V
P
b
b
Figure 3: Assume is an empty sphere passing through a point
and intersecting the
supporting plane of in a circle of radius greater than . Then, contains an empty disk
centered on .
P
Q R T2
V
Lemma 6 Let be a sample point in the -regular zone of . Let be the supporting plane of the
facet through . Any empty sphere passing through intersects in a circle whose radius is less
than .
V
V
Proof. The proof is by contradiction. Let be the supporting plane of . Consider an empty sphere
passing through and intersecting along a circle of radius greater than (see Figure 3). Let
be the center of this circle. Let be the point on the segment at distance from . Because
belongs to the -regular zone of ,
. The empty sphere encloses the disk
.
Therefore,
is an empty disk of , centered on and of radius , which contradicts our
assumption.
D
P
Q? R 2
2
V
D
2
V
4 Counting Delaunay edges
P
Q R T2
be a -sample of . The Delaunay triangulation of
Let be a polyhedral surface and
connects two points iff there exists an empty sphere passing through and . The edge
connecting and is called a Delaunay edge. We will also say that and are Delaunay neighbours.
The number of edges and the number of tetrahedra incident to a vertex lying in the interior
of the convex hull of are related by Euler formula
R
R
RR n° 4453
R A@ R !
10
Attali & Boissonnat
since the boundary of those tetrahedra is a simplicial polyhedron of genus 0. Using the same argument, if lies on the boundary of the convex hull, we have:
By summing over the
we get
R @ R !
vertices, and observing that a tetrahedron has four vertices and an edge two,
! To bound the complexity of the Delaunay triangulation, it is therefore sufficient to count the Delaunay edges of .
We distinguish three types of Delaunay edges : those with both endpoints in the -regular zone,
those with both endpoints in the -singular zone and those with an endpoint in the -regular zone
and the other in the -singular zone. They are counted separately in the following subsections,
We denote by
the set of sample points in the -singular zone of .
4.1 Delaunay edges with both endpoints in the
-regular zone
In this section, we count the Delaunay edges joining two points in the -regular zone.
Lemma 7 Let be a sample point in the -regular zone and most Delaunay neighbours in .
the facet that contains .
has at
Proof. By Lemma 6, any empty sphere passing through intersects in a circle whose radius is
less than . Therefore, the Delaunay neighbours of on are at distance at most from . By
assumption, the disk centered at with radius contains at most points of .
@
SB ^
SB
Lemma 8 Let be a sample point in the -regular zone of a facet . Let of . has at most Delaunay neighbours in the -regular zone of facet .
VB
V
@
be another facet
B 2B
2B P
N N/B
B 2
,2 B
B 2 #2, B
N TN ON B D
2,B
B 2
DS ] N %\N N B
D
N # N B
Proof. Refer to Figure 4.
and
are the supporting planes of and ,
is a Delaunay
neighbour of in the -regular zone of and is an empty sphere passing through and .
intersects the planes and
along two circles whose radii are respectively and . By Lemma 6,
and
.
Let
be the bisector plane of and . Let and be the points symmetric to and with
respect to . Consider the smallest sphere passing through , , and . This sphere intersects
and
in two disks of the same radius
. We claim that
. Indeed, let be
(resp.
) be the bisector plane of and (resp. of and ).
the circumcenter of , and
Observe that
and
. If
,
and the claim is
proved. Otherwise, must belong to one of the two open halfspaces limited by
. If belongs to
the halfspace that contains , encloses and therefore
while in the second it encloses
and
.
N#
2
V
VB
N NB#
V
VHB
V
CB
VHB
N P F ] DS D
B P
B
# N
P
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Complexity of Delaunay triangulations
11
B !*2 B
Figure 4: Any sphere passing through
whose diameter is at least
.
and
2B intersects one of the two planes V
or
VHB in a circle
B,!*2 B N # N ON B # @
and consequently:
B ! 2 B # @ 8
The Delaunay neighbours of in the -regular zone of HB lie in the disk QSR B9@ . This disk
contains at most points of .
Proposition 9 There are at most Delaunay edges with both endpoints in the -regular zone of
.
Proof. The surface has facets. Therefore, by Lemmas 7 and 8, a point in the -regular zone of
has at most Delaunay neighbours. We therefore have :
4.2 Delaunay edges with both endpoints in the
-singular zone
In this section, we count the Delaunay edges joining two points in the -singular zone.
Proposition 10 The number of Delaunay edges with both endpoints in the -singular zone is less
than
Proof. By Proposition 5, the number
RR n° 4453
@
- - of sample points in the -singular zone is at most
-
12
Attali & Boissonnat
Hence, the number of Delaunay edges in the -singular zone is at most
! - .
Figure 5: Example of a Delaunay triangulation of points having a quadratic number of edges.
Even if such a configuration can occur for a subset of the sample points, the number of Delaunay
edges involved in this configuration is .
4.3 Delaunay edges joining the -regular and the -singular zones
In this section, we count the Delaunay edges with one endpoint in the -regular zone and the other
in the -singular zone.
We first introduce a geometric construction of independent interest that will be useful.
Let be a plane and
be a set of points. We assign to each point of
the region
consisting of the points
for which the sphere tangent to at and passing through encloses
no point of
(see Figure 6). In other words, if
denotes the radius of the sphere tangent to
at and passing through , we have:
T X V
"V
V 4O V T 1\ &"V ) 2" 5 # 52 O It is easy to see that the set of all , [ , is a subdivision of V we note (see Figure
9). Let be the paraboloid of revolution with focus and director plane V . The paraboloid consists of the centers of the spheres passing through and tangent to V . Assume that the points
are all located above plane V . If not, we replace by the point symmetric to with respect to
V , which does not change . Let us consider the lower envelope of the collection of paraboloids
. Cell T is the projection of the portion of the lower envelope contributed by (see
V
Figures 6 and 9).
INRIA
Complexity of Delaunay triangulations
13
V
V
T T V
Figure 6: The cell
is the set of contact points between a plane and a sphere passing through
and tangent to . The part of the paraboloid on the lower envelope of the paraboloids projects
.
to the cell
T O2 .
the bisector 452 of 452 , i.e. the points * V such that O TUConsider
O2 is the projection on V of the intersection of the paraboloids and . As easy computations can show, the bisector T452 of and 2 is a circle or a line (considered as a degenerated
circle). Let 4O2 452 \&&V 5 # 52 O Since TUO2 is a circle, TUO2 is either a disk, in which case we rename it Q TUO2 , or the
complementary set of a disk Q T452 . We therefore have
1
T452 ]Q TUO2 Q TUO2 It follows that the edges T452 of T are circle arcs that we call convex or concave wrt depending whether the disk Q TUO2 (whose boundary contains 452 ) is labelled > or ! (see
Figure 7). Observe that the convex edges of T are included in the boundary of the convex hull of
T .
Proposition 11 The number of Delaunay edges with one endpoint in the -regular zone and the
other in the -singular zone is at most :
V
->
b
Proof. Let be a facet of and the supporting plane of . We bound the number of Delaunay
edges with one endpoint in
and the other in
, i.e. the number of Delaunay edges
joining the -singular
zone
and
the
-regular
zone
of
.
We denote by the restriction of the subdivision introduced above to , and, for
,
we denote by
the cell of
associated to .
T RR n° 4453
]
c 14
Attali & Boissonnat
452 2 T 452 2
Figure 7: The bold edges are the convex edges of the shaded cells. The edge
which is
concave wrt is convex wrt . The convex edges of a cell lie on the boundary of its convex hull.
show that the Delaunay neighbours of that belong to the -regular zone of belong to
T We Z first
@ . Consider a Delaunay edge T with , ^ V and W&
] b . Let P be an
empty sphere passing through and , F its center (see Figure 8). By Lemma 6, P intersects V in a
circle whose radius N is less than . For a point D on the segment F8 , we note P the sphere centered
at D and passing through . Because P encloses P , P is an empty sphere. For D F , P intersects
of D on F8 for which
V . For D A , P does not intersect V . Consequently, there exists a position
P is tangent to V .Z Let AP1] V for such a point D . We have & and ! $# @ N # @ .
Hence, @ . Now,Z let us consider a Delaunay edge with 4 ] V . Applying
Lemma leads to T @ .
P
P
F
D
V
Figure 8: Every sphere
tangent to .
P
passing through
and
V
"V
P
contains a sphere passing through
and
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Complexity of Delaunay triangulations
15
Let be the number of Delaunay edges between
is a subdivision of and Proposition 5 :
#
b
and . We have, using the fact that
Z @
# > 9L T Z @ # > @ - J L T Let us bound J L T . Given a cell T , we bound the length of its convex edges. By
summing over all " , all edges in will be taken into account.
X
The convex edges of are contained in the boundary of the convex hull of T . Since T , the length of the boundary of the convex hull of T is at most the length of L . Consequently:
J
9L T # J 9L (
#
- , we have:
Since, by Proposition 5, T #
>
J 9L
By summing over all the facets, we conclude that the total number of Delaunay edges with one
endpoint in the -regular zone and the other in -singular zone is at most :
->
4.4 Main result
We sum up our results in the following theorem :
8
Theorem 12 Let be a polyhedral surface and a -sample of of size
of edges in the Delaunay triangulation
of is at most :
- > @ > -@ . The number
It should be observed that the bound does not depend on the relative position of the facets (provided that their relative interiors do not intersect). Notice also that the bound is not meaningful when
, which is the case of the quadratic example in Figure 5.
a
RR n° 4453
16
Attali & Boissonnat
5 Conclusion
We have shown that, under a mild sampling condition, the Delaunay triangulation of points scattered
over a fixed polyhedral surface or any fixed pure piece-wise linear complex has linear complexity.
Therefore, we (partially) answered an old question of Boissonnat [8]. Our sampling condition does
not involve any randomness (as in the work by Golin and Na [20]) and is less restrictive than Erickson’s one [18].
Although the sampling condition has been expressed in a simple and intuitive way, the linear
bound holds under a more general setting. Indeed, all we need for the proof is to subdivide the
surface in two zones, an -regular zone where one can apply Lemma 6 and an -singular zone
containing points.
As mentionned in the introduction, Erickson has shown that the Delaunay triangulation of
points distributed on a cylinder may be quadratic. To understand where our analysis fails for such
an example, one has to remember that our proof relies on Lemma 6 which states that empty balls
intersect polyhedral surfaces in disks whose area is smaller than , which is not the case anymore
in Erickson’s example.
The main open question is of course to consider the case of smooth surfaces. The lower bound obtained by Erickson for cylinders show that a linear bound does not hold for arbitrary
surfaces. We conjecture that, for generic surfaces, the complexity of the Delaunay triangulation is
still linear. We say that a surface is generic if 1. its maximal balls intersect at a finite number of
so-called contact points, and 2. the intersection of with the union of the maximal balls with only
one contact point form a set of curves of finite length on . In particular, generic surfaces cannot
contain spherical nor cylindrical pieces.
References
[1] N. Amenta, M. Bern, and M. Kamvysselis. A new Voronoi-based surface reconstruction algorithm. In Proc. SIGGRAPH ’98, pages 415–412, July 1998.
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Contents
1 Introduction
2 Definitions and notations
2.1 Voronoi diagrams and Delaunay triangulations .
2.2 Notations . . . . . . . . . . . . . . . . . . . .
2.3 Polyhedral surfaces . . . . . . . . . . . . . . .
2.4 Sample . . . . . . . . . . . . . . . . . . . . .
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3 Preliminary results
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4 Counting Delaunay edges
4.1 Delaunay edges with both endpoints in the
-regular zone . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Delaunay edges with both endpoints in the
-singular zone . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Delaunay edges joining the -regular and the -singular zones
4.4 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Conclusion
RR n° 4453
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Attali & Boissonnat
INRIA
Figure 9: Decomposition of a facet into cells for different set of points . The lower envelope of
has been represented. The red spheres represent the points of
and the
the paraboloid red lines materialize the projection of the points of
on the plane . The bisector of two points is
a circle. The projection of on do not belong necessary to its cell. The decomposition of can
have a quadratic number of edges.
V
V
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